‘Or’ is a troublesome particle in dire need of regimentation. Besides its two disjunctive meanings, the inclusive and the exclusive, there is also what I call the ‘or’ of identity. The inclusive meaning, corresponding to the Latin vel, is illustrated by ‘He is either morally obtuse or intellectually obtuse.’ This allows that the person in question may be both.
The exclusive meaning, corresponding to the Latin aut, is exemplified by the standard menu inscription, ‘soup or salad,’ which means one or the other, but not both. Logicians view the inclusive ‘or’ as a basic propositional connective. Thus our first example would be symbolized by p v q, where p is the proposition expressed by ‘He is morally obtuse’; q the proposition expressed by ‘He is intellectually obtuse’; with ‘v’ -- in honor of vel -- standing for inclusive disjunction. Exclusive ‘or’ can now be defined as follows: p aut q =df p v q & ~(p & q), where the tilde and the ampersand, both propositional connectives, represent negation and conjunction respectively.
How is ‘or’ functioning in this sentence: ‘Philosophers of a realist bent posit facts or states of affairs as truth-makers.’ Clearly, no disjunction is being conveyed. The idea is that facts just are states of affairs, or that ‘fact’ and ‘state of affairs’ are being used interchangeably. Indeed, the preceding sentence exemplifies a use of ‘or’ that does not express a disjunction. Hence, ‘or’ of identity. Such uses of ‘or’ can be replaced by ‘i.e.,’ id est.
Why is this important? Well, it is important if your aim to is write and think with precision and self-awareness. If that is not your aim, then it ought to be.
A point rather orthogonal to your post, but interesting nonetheless:
It's ahistorical to say that aut is logically exclusive. Aut is generally used in situations where the disjuncts are incompatible anyway; the choice of aut versus vel is more of a pragmatic issue than a logical one.
In fact there's a fairly broad swath of the semantics literature devoted to the question of whether natural languages even have exclusive or at all. The consensus appears to be that they do not, but instead rely on pragmatics, facts about the world, or explicit "but not both"-type constructions to differentiate between inclusive and exclusive. It's an interesting topic.
Posted by: A. Cooper | Tuesday, February 24, 2009 at 12:37 PM
Shouldn't there be commas around such a parenthetical 'or'? We're not completely happy with the first term, so we introduce an alternative one "or if you like..." might be the full construction. In that case, I think it's more idiom than a construction of logic.
If it operates on anything it operates on the terms, rather than on the states the terms represent. So you can choose one term or both terms to represent the state if it suits you better.
Posted by: John Cassidy | Tuesday, February 24, 2009 at 02:51 PM
A Finnish woman complained to me about the third sense of "or" (the one which is sometimes expressible by "i.e.") and that's when I started using "a.k.a." oftener.
I liked Quine's idea of a word "exclor" for exclusive disjunction, but that sort of thing never catches on. I'd prefer "twexclor" to reinforce the binary aspect, since exclusive "or" doesn't behave like non-exclusive "or" in longer sequences.
Could never go along, though, with Quine on the notion that "only if" means "if...then."
Posted by: Ben Udell | Wednesday, February 25, 2009 at 07:48 PM
It's simple!
Let "aor" denote non-exclusive disjunctions (short for "and/or").
Let "orr" denote exclusive disjunctions (following the idea of "iff").
Let "aka" denote alternative expressions.
There! No need for this "exclor" nonsense. Wasn't he one of the evil Transformers?
Posted by: D Bnonn Tennant | Thursday, February 26, 2009 at 12:28 AM
I usually write OR and XOR, following the electrical engineering terminology . . .
Posted by: A. Cooper | Saturday, February 28, 2009 at 02:33 PM